pb makes some really good points--in our case DS phobias were(are) part and parcel of his second E.

DD is more like the "anxious personality" and seems like she was born that way. There is actually a commonly studied gene variant associated with both higher levels of anxiety and some aspects of intelligence.
https://www.snpedia.com/index.php/Rs4680
(We know DS has this variant, as do I, so DD is at least heterozygous. I know this isn't what you asked about, but it's really interesting to see that some of us are just made that way!)

Here's an interesting article about the arachnophobia: http://www.iflscience.com/plants-and-animals/our-fear-spiders-innate-not-learned

(I know that is more than a little OT but it's interesting to me that there are so many shared fears across cultures. I wonder how those costume characters relate? Maybe something about the size, exaggerated features?)

And here is a really cool book that I like to use with bright/verbal children to encourage them they can shape their experience:

http://www.amazon.com/Fantastic-Ela...rect=true&ref_=oh_aui_detailpage_o01_s00

No matter the anxiety triggers, developing some coping skills won't hurt anything.